Graph Homomorphisms Based On Particular Total Colorings of Graphs and Graphic Lattices Bing Yao1, Hongyu Wang2;y 1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070 CHINA 2. School of Electronics Engineering and Computer Science, Peking University, Beijing, 100871, CHINA y Corresponding author Hongyu Wang's email: [email protected] Abstract Lattice-based cryptography is not only for thwarting future quantum computers, and is also the basis of Fully Homomorphic Encryption. Motivated from the advantage of graph homomor- phisms we combine graph homomorphisms with graph total colorings together for designing new types of graph homomorphisms: totally-colored graph homomorphisms, graphic-lattice homo- morphisms from sets to sets, every-zero graphic group homomorphisms from sets to sets. Our graph-homomorphism lattices are made up by graph homomorphisms. These new homomor- phisms induce some problems of graph theory, for example, Number String Decomposition and Graph Homomorphism Problem. Key words: Graph homomorphism; graphic lattice; total coloring; isomorphism; lattice- based cryptography; topological coding. 1 Introduction and preliminary A new security method built on an underlying architecture known as lattice-based cryptography hides data inside complex mathematical problems. Lattice-based cryptography is not only for thwarting future quantum computers, and is also the basis of another encryption technology, called Fully Homomorphic Encryption, which could make it possible to perform calculations on a file without ever seeing sensitive data or exposing it to hackers (Ref. [1, 2, 3]). arXiv:2005.02279v1 [math.CO] 5 May 2020 1.1 Graph homomorphisms in Homomorphic Encryption Homomorphisms provide a way of simplifying the structure of objects one wishes to study while preserving much of it that is of significance. It is not surprising that homomorphisms also appeared in graph theory, and that they have proven useful in many areas (Ref. [1, 3]). Graph homomor- phisms have a great deal of applications in graph theory, computer science and other fields. The connection between locally constrained graph homomorphisms and degree matrices arising from an equitable partition of a graph have been explored in [8]. A main computational issue is: for every graph H classifying the decision problem whether an input graph G has a homomorphism of given type to the fixed graph H as either NP-complete or polynomially solvable (Ref. [2]). The comprehensive survey by Zhu [9] contains many other intriguing problems about graph homomorphism. We, in this article, try to provide some design of graph homomorphisms which are based on topological structures and graph colorings. Topsnut-gpws (Ref. [16, 17]) will play main roles in our homomorphisms (six colored graphs (a)-(f) shown in Fig.2 are the Topsnut-gpws), because Topsnut- gpws are made up of two kinds of mathematical objects: topological structure and algebraic relation, such that attacker switch back and forth in two different languages, and are unable to convey useful information. 1.2 Definition for graph homomorphisms We will use the standard notation and terminology of graph theory in this paper. Graphs will be simple, loopless and finite. A (p; q)-graph is a graph having p vertices and q edges. The cardinality of a set X is denoted as jXj, so the degree of a vertex x in a (p; q)-graph G is written as degG(x) = jN(x)j, where N(x) is the set of neighbors of the vertex x. A vertex y is called a leaf if degG(y) = 1. A symbol [a; b] stands for an integer set fa; a + 1; : : : ; bg with two integers a; b subject to a < b. All non-negative integers are collected in the set Z0. A graph G admits a labelling f : V (G) ! [a; b] means that f(x) 6= f(y) for any pair of distinct vertices x; y 2 V (G) and, admits a coloring g : V (G) ! [a; b] means that g(u) = g(v) for some two distinct vertices u; v 2 V (G). For a mapping f : S ⊂ V (G) [ E(G) ! [1;M], write color set by f(S) = ff(w): w 2 Sg. The definition of a graph homomorphism is shown as follows: Definition 1. [6] A graph homomorphism G ! H from a graph G into another graph H is a mapping f : V (G) ! V (H) such that f(u)f(v) 2 E(H) for each edge uv 2 E(G). (see examples shown in Fig.1.) 2 1 3 1 1 3 2 1 2 3 3 2 3 2 H G1 G2 Figure 1: Two graph homomorphisms Gi ! H based on θi : Gi ! H for i = 1; 2. Remark 1. By [2], we have the following concepts: (a) A homomorphism from a graph G to itself is called an endomorphism. An isomorphism from G to H is a particularly graph homomorphism from G to H, also, they are homomorphically equivalent. 2 (b) Two graphs are homomorphically equivalent if each admits a homomorphism to the other, denoted as G $ H which contains a homomorphism G ! H from G to H, and another homomor- phism H ! G from H to G. (c) A homomorphism to the complete graph Kn is exactly an n-coloring, so a homomorphism of G to H is also called an H-coloring of G. The homomorphism problem for a fixed graph H, also called the H-coloring problem, asks whether or not an input graph G admits a homomorphism to H. (d) By analogy with classical colorings, we associate with each H-coloring f of G a partition −1 of V (G) into the sets Sh = f (h), h 2 V (H). It is clear that a mapping f : V (G) ! V (H) is a homomorphism of G to H if and only if the associated partition satisfies the following two constraints: (a-1) if hh is not a loop in H, then the set Sh is independent in G; and 0 (a-2) if hh is not an edge (arc) of H, then there are no edges (arcs) from the set Sh to the set Sh0 in G. Thus for a graph G to admit an H-coloring is equivalent to admitting a partition satisfying (a-1) and (a-2). (e) If H,H0 are homomorphically equivalent, then a graph G is H-colorable if and only if it is H0-colorable. (f) Suppose that H is a subgraph of G. We say that G retracts to H, if there exists a homomorphism f : G ! H, called a retraction, such that f(u) = u for any vertex of H.A core is a graph which does not retract to a proper subgraph. Any graph is homomorphically equivalent to a core. 2 Results on graph homomorphisms 2.1 Graph homomorphisms of uncolored graphs By Definition 1, we have a result as follows: Proposition 1. Suppose that ' : G ! H is a graph homomorphism. Then ' is an isomorphism if and only if ' is bijective and also a homomorphism. In particular, if G = H then ' is an automorphism if and only if it is bijective. Definition 2. [1] A graph homomorphism ' : G ! H is called faithful if '(G) is an induced subgraph of H, and called full if uv 2 E(G) if and only if '(u)'(v) 2 E(H). A graph homomorphism ' : G ! H is faithful when there is an edge between any two pre- images (inverse image) '−1(x) and '−1(y) such that xy is an edge of H, '−1(x) [ '−1(y) induces a complete bipartite graph whenever xy 2 E(H). Moreover, '−1(u)'−1(v) is an edge in G if and only if uv is an edge in H, thus Theorem 2. [1] A faithful bijective graph homomorphism is an isomorphism, that is G =∼ H. 3 ∗ ∗ ∗ ∗ Theorem 3. There are infinite graphs Gn forming a sequence fGng, such that Gn ! Gn−1 is really a graph homomorphism for n ≥ 1. Proof. First, we present an algorithm as follows: G0 is a triangle ∆x1x2x3, we use a coloring h to color the vertices of G0 as h(xi) = 0 with i 2 [1; 3]. Step 1: Add a new y vertex for each edge xixj of the triangle ∆x1x2x3 with i 6= j, and join y with two vertices xi and xj of the edge xixj by two new edges yxi and yxj, the resulting graph is denoted by G1, and color y with h(y) = 1. Step 2: Add a new w vertex for each edge uv of G1 if h(u) = 1 and h(v) = 0 (or h(v) = 1 and h(u) = 0), and join y respectively with two vertices u and v by two new edges wu and wv, the resulting graph is denoted by G2, and color w with h(w) = 2. Step n: Add a new γ vertex for each edge αβ of Gn−1 if h(α) = n − 1 and h(β) = n − 2 (or h(α) = n − 2 and h(β) = n − 1), and join γ respectively with two vertices α and β by two new edges γα and γβ, the resulting graph is denoted by Gn, and color γ with h(γ) = n. ∗ Second, we construct a graph Gn = Gn [ K1 with n ≥ 0, where K1 is a complete graph ∗ ∗ of one vertex z0. For each n ≥ 1, there is a mapping θn : V (Gn) ! V (Gn−1) in this way: ∗ n ∗ n n V (Gn n V(2)) = V (Gn−1 n V (K1)), each x 2 V(2) holds θn(x) = z0, where V(2) is the set of vertices ∗ ∗ ∗ of degree two in Gn. So Gn ! Gn−1 is really a graph homomorphism.